Algebra 2 B - Exponential/Logarithmic Functions

Questions and Answers

What is the definition of Lesson 6?

Linear Functions

Exponential Equations (correct)

Logarithmic Functions

Quadratic Equations

Solve for x: $4^{3x} = 4^{x-1}$.

-1/2

Solve for x: $3^{5x+7} = 3^{7x-1}$.

Solve $9^{4x} = 243$ for x.

5/8 Signup and view all the answers

Solve for x: $64 = 2^{2x-4}$.

5 Signup and view all the answers

Solve for x: $5^{2x-5} = 125$.

4 Signup and view all the answers

Solve for x: $4 imes 2^x = 32$.

3 Signup and view all the answers

Solve $100^{4x} = 1000$ for x.

3/8 Signup and view all the answers

Solve for x: $e^{2x} = 4$.

ln 4 / 2 Signup and view all the answers

Solve for x: $5^{-2x-9} = 5^{4x+15}$.

-4 Signup and view all the answers

Solve for x: $9^{2x} = 27$.

3/4 Signup and view all the answers

Solve for x: $8^{7x-5} = 8^{4x-3}$.

2/3 Signup and view all the answers

Solve for x: $10^{3x} = 12$.

log 12 / 3 Signup and view all the answers

Solve for x: $128 = 2^{3x-5}$.

4 Signup and view all the answers

If $7^{6x-11} = 7^{2x+1}$, what is the value of x?

3 Signup and view all the answers

Solve for x: $8 imes e^{x-1} = 56$.

ln 7 + 1 Signup and view all the answers

Solve for x: $2^{4x} = 6$.

log_2 6 / 4 Signup and view all the answers

Solve for x: $16^{3x} = 64$.

1/2 Signup and view all the answers

What is the definition of Lesson 7?

Properties of Logarithms Signup and view all the answers

Which expression is equivalent to $log_4 x^5$?

5 log_4 x Signup and view all the answers

Match each logarithm with its equivalent expression.

log_7 y^{10} = 10 log_7 y ln 7^{10} = 10 ln 7 log 7^4 = 4 log 7 Signup and view all the answers

Which expressions are equivalent to $log_9 14$?

A and B Signup and view all the answers

What is the value of $log_{27} 165$?

1.549 Signup and view all the answers

What is the value of $log_6 49$?

2.17 Signup and view all the answers

Which expression is equivalent to $log_2 6^x$?

x log_2 6 Signup and view all the answers

What is the value of $log_{17} 2$?

0.24 Signup and view all the answers

What is the value of $log_5 75$?

2.68 Signup and view all the answers

Which expression is equivalent to $log_3 24^2$?

2 log_3 24 Signup and view all the answers

What is the definition of Lesson 8?

How to Solve Exponential Equations with Technology Signup and view all the answers

If $4^{3x} = 37.9$, what is the value of x?

x≈0.874 Signup and view all the answers

Solve for x: $5^{x/2} = 65$, round your answer to three places.

5.187 Signup and view all the answers

Solve $6^{3x+1} = 10.8$ for x. Which answer shows the correct steps?

x≈0.109 Signup and view all the answers

Solve the equation for x: $8^{x+2} = 99.6$.

x≈0.213 Signup and view all the answers

If $3^{(x/2)} + 4 = 7.32$, what is the value of x?

x≈−4.38 Signup and view all the answers

If $17^{2x-3} = 35$, what is the value of x? Round your answer to three places.

2.127 Signup and view all the answers

Solve for x: $2^{(x/3)} = 100.6$.

x≈19.96 Signup and view all the answers

If $3^{(x/4)} + 2 = 56.8$, what is the value of x?

x≈6.708 Signup and view all the answers

Solve $25^{2x-1} = 400$ for x. Which answer shows the correct steps?

x≈1.431 Signup and view all the answers

If $6^{4x} = 82$, what is the value of x?

0.615 Signup and view all the answers

What is the definition of Lesson 9?

Exponentials in Context Signup and view all the answers

On the first day of a social media ad campaign, a website had 15 unique visitors. Over the following several days, the website received three times the number of unique visitors as the previous day. Which equation can be used to determine the number of unique visitors, v, that the website would have on day d of the ad campaign?

v=15⋅3^{(d-1)} Signup and view all the answers

The membership fee at a sports club was $600 per year. Each year, the membership fee increased by 7%. Which equation models the cost of the membership fee, y, after x years?

y=600(1.07)^x Signup and view all the answers

A nearby pond has 5 frogs, and the population doubles every year. Which inequality, where t is the number of years, models when the population of frogs will be greater than 75?

5(2)^t > 75 Signup and view all the answers

On which day of the ad campaign did the website have 10,000 visitors based on the model $v=15⋅3^{(d-1)}$?

7 Signup and view all the answers

Study Notes

Exponential Equations

An exponential equation can take the form ( a^x = b ), where ( a ) and ( b ) are constants.
Example: For ( 4^{3x} = 4^{x - 1} ), solving yields ( x = -\frac{1}{2} ).

Solving for x in Various Equations

( 3^{5x + 7} = 3^{7x - 1} ) results in ( x = 4 ).
For ( 9^{4x} = 243 ), the solution is ( x = \frac{5}{8} ).
The equation ( 64 = 2^{2x - 4} ) gives ( x = 5 ).
In ( 5^{2x - 5} = 125 ), ( x ) equals 4.
The equation ( 4 \cdot 2^x = 32 ) solves to ( x = 3 ).
( 100^{4x} = 1000 ) yields ( x = \frac{3}{8} ).
The equation ( e^{2x} = 4 ) resolves to ( x = \frac{\ln 4}{2} ).
For ( 5^{-2x - 9} = 5^{4x + 15} ), the solution is ( x = -4 ).
Solving ( 9^{2x} = 27 ) gives ( x = \frac{3}{4} ).
Equation ( 8^{7x - 5} = 8^{4x - 3} ) results in ( x = \frac{2}{3} ).
( 10^{3x} = 12 ) resolves to ( x = \frac{\log 12}{3} ).
For ( 128 = 2^{3x - 5} ), the answer is ( x = 4 ).
If ( 7^{6x - 11} = 7^{2x + 1} ), then ( x = 3 ).
Solving ( 8 \cdot e^{x - 1} = 56 ) yields ( x = \ln 7 + 1 ).
The equation ( 2^{4x} = 6 ) simplifies to ( x = \frac{\log_2 6}{4} ).
For ( 16^{3x} = 64 ), the solution is ( x = \frac{1}{2} ).

Properties of Logarithms

Logarithmic expressions can be manipulated using properties:
- ( \log_a (x^b) = b \cdot \log_a (x) ).
Example: ( \log_4 (x^5) ) is equivalent to ( 5 \log_4 (x) ).
( \log_7 (y^{10}) ) equals ( 10 \log_7 (y) ).
Natural logarithms: ( \ln (7^{10}) = 10 \ln (7) ).
( \log (7^4) ) can be expressed as ( 4 \log (7) ).

Evaluating Logarithms

( \log_9 (14) ) can be calculated as ( \frac{\ln 14}{\ln 9} ) or ( \frac{\log 14}{\log 9} ).
Calculated values include ( \log_{27} (165) \approx 1.549 ) and ( \log_6 (49) \approx 2.17 ).
Expression ( \log_2 (6^x) = x \log_2 (6) ).
Other calculated values: ( \log_{17} (2) \approx 0.24 ) and ( \log_5 (75) \approx 2.68 ).
( \log_3 (24^2) ) simplifies to ( 2 \log_3 (24) ).

Solving Exponential Equations with Technology

Utilizing technology can aid in solving complex exponential equations.
Example: For ( 4^{3x} = 37.9 ), ( x \approx 0.874 ).
( 5^{x/2} = 65 ) results in ( x \approx 5.187 ).
Equation ( 6^{3x + 1} = 10.8 ) yields ( x \approx 0.109 ).
If ( 8^{x + 2} = 99.6 ), then ( x \approx 0.213 ).
For ( 3^{(x/2)} + 4 = 7.32 ), the solution is ( x \approx -4.38 ).
Equation ( 17^{2x - 3} = 35 ) results in ( x \approx 2.127 ).
( 2^{(x/3)} = 100.6 ) leads to ( x \approx 19.96 ).
Solving ( 3^{(x/4)} + 2 = 56.8 ) gives ( x \approx 6.708 ).
The equation ( 25^{(2x - 1)} = 400 ) resolves to ( x \approx 1.431 ).
For ( 6^{4x} = 82 ), the solution is ( x \approx 0.615 ).

Exponentials in Context

Example of a social media ad campaign: Initial visitors are 15, with growth modeled by ( v = 15 \cdot 3^{(d-1)} ).
Membership fee at a sports club initiates at $600, increasing by 7% each year, modeled by ( y = 600(1.07)^x ).
A frog population starts with 5 and doubles each year; the model ( 5(2)^t > 75 ) determines time until the population exceeds 75.
Identifying when the website reaches 10,000 visitors: Day 7 in the campaign.
Cost equation for sports club membership demonstrates percentage increase over time.

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Test your knowledge on exponential equations and their solutions with this flashcard quiz from Algebra 2 B, Unit 2. Perfect for mastering exponential and logarithmic functions, this quiz helps reinforce important concepts and problem-solving skills.